#493062

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,10,16,12,19,30,18)(7,20,23,46,25,47,52,35)(8,14,28,40,26,38,56,42)(9,17,11,15,31,43,32,44)(13,24,34,53,37,54,36,22)(21,45,50,65,48,64,67,60)(27,41,29,39,55,63,57,62)(33,51,59,66,58,68,61,49);(2,4)(5,7,11,24,10,23,31,53,12,25,32,54,30,52,9,22)(6,13,17,35,18,36,44,47,19,37,43,46,16,34,15,20)(8,21,29,51,28,50,55,66,26,48,57,68,56,67,27,49)(14,33,41,60,42,61,62,64,38,58,63,65,40,59,39,45);(5,8,10,28,12,26,30,56)(6,14,16,40,19,38,18,42)(7,21,23,50,25,48,52,67)(9,27,11,29,31,55,32,57)(13,33,34,59,37,58,36,61)(15,39,43,63,44,62,17,41)(20,45,46,65,47,64,35,60)(22,49,24,51,53,66,54,68);(8,26)(14,38)(21,48)(27,55)(28,56)(29,57)(33,58)(39,62)(40,42)(41,63)(45,64)(49,66)(50,67)(51,68)(59,61)(60,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=ghi,h^a=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400258);